Direct Determination of reciprocal coordinate system

The following coordinate system is given:

$a=4\mathrm{.}500$

$b=6\mathrm{.}250$

$c=5\mathrm{.}780$

$=94\mathrm{.}25$

$=112\mathrm{.}74$

$=102\mathrm{.}43$

Determine the volume of the parallelepiped spanned by the three vectors by determining the metric matrix, then calculate its determinant.

Determine the reciprocal coordinate system. Use the metric matrix calculated in the previous step, and invert it to obtain the reciprocal metric matrix.

For this problem, it is not necessary to describe these vectors in an unitary coordinate system.